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   <h4 class="subsectionHead"><span class="titlemark">2.13   </span> <a 
 id="x18-620002.13"></a>Special elements</h4>
<!--l. 1481--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.13.1   </span> <a 
 id="x18-630002.13.1"></a>LumpedMass element</h5>
<!--l. 1482--><p class="noindent" >This element, defined by a single node, allows to introduce additional concentrated mass and/or rotational inertias in
a node. A different mass and rotary inertia may be assigned to each coordinate direction. At present,
individual mass/inertia components can be specified for every degree of freedom of element node. Only
displacement and rotational degrees od freedom are considered. The element features are summarized in
Table&#x00A0;<a 
href="#x18-6300146">46<!--tex4ht:ref: LumpedMasssummary --></a>.
<div class="center" 
>
<!--l. 1485--><p class="noindent" >
<div class="table">
<!--l. 1485--><p class="noindent" ><a 
 id="x18-6300146"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-48" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-48-1g"><col 
id="TBL-48-1"></colgroup><colgroup id="TBL-48-2g"><col 
id="TBL-48-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-48-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-48-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-48-1-2"  
class="td11"><!--l. 1485--><p class="noindent" ><span 
class="cmbx-10">LumpedMass</span>                                                 </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-48-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-48-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-48-2-2"  
class="td11"><!--l. 1485--><p class="noindent" >Lumped mass element                                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-48-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-48-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-48-3-2"  
class="td11"><!--l. 1485--><p class="noindent" ><span 
class="cmtt-10">components</span>&#x00A0;#<span 
class="cmr-5">(ra)</span>                                                                            </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-48-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-48-4-1"  
class="td11">Parameters           </td><td  style="white-space:wrap; text-align:left;" id="TBL-48-4-2"  
class="td11"><!--l. 1486--><p class="noindent" ><span 
class="cmtt-10">components</span>: allows to specify additional concentrated mass
components  (Force*Time<sup><span 
class="cmr-7">2</span></sup>/Length)  and  rotary  inertias
(Force*Length*Time<sup><span 
class="cmr-7">2</span></sup>)  about  the  nodal  coordinate  axes.
Only displacement and rotational degrees of freedom are
considered. The individual DOFs are ordered according to
following rule: first, displacement degrees of freedom in the
direction of x,y, and z axes (if any), followed by rotational
DOFs around x,y, and z axes (if any).                           </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-48-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-48-5-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-48-5-2"  
class="td11"><!--l. 1487--><p class="noindent" >Only displacement and roational DOFs are considered (<span 
class="cmmi-10">D</span><sub><span 
class="cmmi-7">u</span></sub>,
<span 
class="cmmi-10">D</span><sub><span 
class="cmmi-7">v</span></sub>, <span 
class="cmmi-10">D</span><sub><span 
class="cmmi-7">w</span></sub>, <span 
class="cmmi-10">R</span><sub><span 
class="cmmi-7">u</span></sub>, <span 
class="cmmi-10">R</span><sub><span 
class="cmmi-7">v</span></sub>, and <span 
class="cmmi-10">R</span><sub><span 
class="cmmi-7">w</span></sub>).                                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-48-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-48-6-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-48-6-2"  
class="td11"><!--l. 1488--><p class="noindent" >-                                                                            </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-48-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-48-7-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-48-7-2"  
class="td11"><!--l. 1489--><p class="noindent" >-                                                                            </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-48-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-48-8-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-48-8-2"  
class="td11"><!--l. 1490--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-48-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-48-9-1"  
class="td11">CS properties </td> <td  style="white-space:wrap; text-align:left;" id="TBL-48-9-2"  
class="td11"><!--l. 1491--><p class="noindent" >-</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-48-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-48-10-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-48-10-2"  
class="td11"><!--l. 1492--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-48-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-48-11-1"  
class="td11">Status </td> <td  style="white-space:wrap; text-align:left;" id="TBL-48-11-2"  
class="td11"><!--l. 1493--><p class="noindent" >Reliable</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-48-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-48-12-1"  
class="td11">                </td></tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-48-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-48-13-1"  
class="td11"></td></tr></table>
</div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;46: </span><span  
class="content">LumpedMass element summary</span></div><!--tex4ht:label?: x18-6300146 -->
</div><hr class="endfloat" />
</div>
</div>
<!--l. 1496--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.13.2   </span> <a 
 id="x18-640002.13.2"></a>Spring element</h5>
<!--l. 1497--><p class="noindent" >This element represent longitudial or torsional spring element. It is defined by two nodes, orientation and a spring
constant. The spring element has no mass associated, the mass can be added using LumpedMass element. The spring
is linear and works the same way in tension or in compression. The element features are summarized in
Table&#x00A0;<a 
href="#x18-6400147">47<!--tex4ht:ref: Springsummary --></a>.
<div class="center" 
>
<!--l. 1500--><p class="noindent" >
                                                                                           
                                                                                           
<div class="table">
<!--l. 1500--><p class="noindent" ><a 
 id="x18-6400147"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-49" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-49-1g"><col 
id="TBL-49-1"></colgroup><colgroup id="TBL-49-2g"><col 
id="TBL-49-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-49-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-49-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-49-1-2"  
class="td11"><!--l. 1500--><p class="noindent" ><span 
class="cmbx-10">Spring</span>                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-49-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-49-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-49-2-2"  
class="td11"><!--l. 1500--><p class="noindent" >Spring element                                                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-49-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-49-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-49-3-2"  
class="td11"><!--l. 1500--><p class="noindent" ><span 
class="cmtt-10">mode</span>&#x00A0;#<span 
class="cmr-5">(in) </span><span 
class="cmtt-10">k</span>&#x00A0;#<span 
class="cmr-5">(rn) </span>[<span 
class="cmtt-10">m</span>&#x00A0;#<span 
class="cmr-5">(rn)</span>] <span 
class="cmtt-10">orientation</span>&#x00A0;#<span 
class="cmr-5">(ra)</span>                     </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-49-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-49-4-1"  
class="td11">Parameters           </td><td  style="white-space:wrap; text-align:left;" id="TBL-49-4-2"  
class="td11"><!--l. 1501--><p class="noindent" ><span 
class="cmtt-10">mode</span>: defines the type of spring element (see Table&#x00A0;<a 
href="#x18-6400248">48<!--tex4ht:ref: spring_mode_table --></a>).    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-49-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-49-5-1"  
class="td11">                </td><td  style="white-space:wrap; text-align:left;" id="TBL-49-5-2"  
class="td11"><!--l. 1502--><p class="noindent" ><span 
class="cmtt-10">k</span>: determines the spring constant, corresponding units are
[Force/Length]      for      longitudinal      spring      and
[Force*Length/Radian] for torsional spring.                    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-49-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-49-6-1"  
class="td11">                </td><td  style="white-space:wrap; text-align:left;" id="TBL-49-6-2"  
class="td11"><!--l. 1503--><p class="noindent" ><span 
class="cmtt-10">orientation</span>:defines orientation vector of spring element
(of size 3) - for longitudinal spring it defines the direction
of spring, for torsional spring it defines the axis of rotation.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-49-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-49-7-1"  
class="td11">                </td><td  style="white-space:wrap; text-align:left;" id="TBL-49-7-2"  
class="td11"><!--l. 1504--><p class="noindent" ><span 
class="cmtt-10">m</span>:  determines  optional  mass  of  the  element,  zero  value
assumed by default.                                                   </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-49-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-49-8-1"  
class="td11">Note                   </td><td  style="white-space:wrap; text-align:left;" id="TBL-49-8-2"  
class="td11"><!--l. 1505--><p class="noindent" >the spring element nodes doesn&#8217;t need to be coincident,
but  the  spring  orientation  is  always  determined  by
<span 
class="cmtt-10">orientation </span>vector.                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-49-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-49-9-1"  
class="td11">                </td></tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-49-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-49-10-1"  
class="td11"></td></tr></table>
</div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;47: </span><span  
class="content">Spring element summary</span></div><!--tex4ht:label?: x18-6400147 -->
</div><hr class="endfloat" />
</div>
</div>
   <div class="table">
                                                                                           
                                                                                           
<!--l. 1509--><p class="indent" >   <a 
 id="x18-6400248"></a><hr class="float"><div class="float" 
>
                                                                                           
                                                                                           
<div class="tabular"> <table id="TBL-50" class="tabular" 
cellspacing="0" cellpadding="0"  
><colgroup id="TBL-50-1g"><col 
id="TBL-50-1"><col 
id="TBL-50-2"><col 
id="TBL-50-3"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-50-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-1-1"  
class="td11">mode</td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-1-2"  
class="td11">description                                                                  </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-50-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-2-1"  
class="td11">0       </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-2-2"  
class="td11">1D spring element along x-axis,                                       </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-50-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-3-1"  
class="td11">     </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-3-2"  
class="td11">requires D_u DOF in each node, orientation vector is <span 
class="cmsy-10">{</span>1,0,0<span 
class="cmsy-10">}</span></td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-50-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-4-1"  
class="td11">1       </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-4-2"  
class="td11">2D spring element in xy plane,                                        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-50-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-5-1"  
class="td11">     </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-5-2"  
class="td11">requires D_u and D_v DOFs in each node                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-50-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-6-1"  
class="td11">     </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-6-2"  
class="td11">(orientation vector should be in xy plane)                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-50-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-7-1"  
class="td11">2       </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-7-2"  
class="td11">2D spring element in xz plane,                                        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-50-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-8-1"  
class="td11">     </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-8-2"  
class="td11">requires D_u and D_w DOFs in each node                        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-50-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-9-1"  
class="td11">     </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-9-2"  
class="td11">(orientation vector should be in xz plane)                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-50-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-10-1"  
class="td11">3       </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-10-2"  
class="td11">2D torsional spring element in xz plane,                            </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-50-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-11-1"  
class="td11">     </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-11-2"  
class="td11">requires R_v DOFs in each node                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-50-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-12-1"  
class="td11">4       </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-12-2"  
class="td11">3D spring element in space,                                            </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-50-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-13-1"  
class="td11">     </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-13-2"  
class="td11">requires D_u, D_v, and D_w DOFs in each node                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-50-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-14-1"  
class="td11">5       </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-14-2"  
class="td11">3D torsional spring in space,                                           </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-50-15-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-15-1"  
class="td11">     </td><td  style="white-space:nowrap; text-align:left;" id="TBL-50-15-2"  
class="td11">requires R_u, R_v, and R_w DOFs in each node                </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-50-16-"><td  style="white-space:nowrap; text-align:left;" id="TBL-50-16-1"  
class="td11">     </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;48: </span><span  
class="content">Supported spring element modes </span></div><!--tex4ht:label?: x18-6400248 -->
                                                                                           
                                                                                           
   </div><hr class="endfloat" />
   </div>
                                                                                           
                                                                                           
                                                                                           
                                                                                           
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